Why is it that you swap the phone to the other ear when the conversation suddenly gets important? Maybe you haven't ever noticed this. Maybe you will now!

Before answering the question¹, lets start with some general rules²:

Logic and Emotion

The left and right sides of the brain are used for different tasks:
Rule 1. The left hemisphere of the brain is generally used for logic and language.
Rule 2. The right hemisphere of the brain is generally used for emotion and expression.

So, the left side of your brain prefers order, whereas the right side of your brain prefers chaos.

Girls and Boys

It is suggested that there is a difference between women and men when it comes to brain hemisphere dominance:
Rule 3. Men are generally left-side dominant.
Rule 4. Women are generally right-side dominant.

So, when presented with a problem, a woman's reaction will be "how can I make you feel better?" whereas the men will say "how can I fix it?"

It is also suggested that women are more able to use both sides of the brain simultaneously:
Rule 5. Men are generally left-side dominant, and that's it.
Rule 6. Women are generally right-side dominant, but with a bit of left-side thrown in for good measure.

So, when presented with a rabbit to kill, a man will go out and kill it, whereas a woman will keep hold of the baby whilst throwing stones at the rabbit.

Left and Right

Each side of the brain accepts sensory input from the opposite side of the body, and also controls motor functions (ie, movement) of the opposite side of the body:
Rule 7. The left side of the brain connects with the right side of the body.
Rule 8. The right side of the brain connects with the left side of the body.

So, when a blackbird appears in a bush on your right, you will be more quickly able to say what it is you saw, whereas when it appears on your left, you will be more quickly able to point at it with your left hand.

Although left- or right-hemisphere dominance is related to whether a person is left- or right-handed, it isn't entirely related:
Rule 9. Left-handed people are quite often right-side dominant.
Rule 10. Right-side dominant people aren't necessarily left-handed.

So, all blackbirds are black birds, but not all black birds are blackbirds.

It should also be noted that:
Rule 11. About 90% of people are right-handed.
Rule 12. About 10% of people are left-handed.
And:
Rule 13. Less than 1% of people are ambidextrous.

So, all blackbirds are black birds, except the albino ones.

'Honey, I've Crashed the Car'

So, back - finally - to the phone call.

Right-handed men

If you are a right-handed man, the chances are, you will pick up the phone with your right hand and hold it to your right ear ^rule11. You will have been doing this all your life, so your brain will have trained itself to deal with this even though your right ear isn't quite as good at listening as your left ear ^rule2. However, when the conversation gets really important (ie, the wife mentions that she's crashed the car), you unconsciously swap to you left ear, which is a few inches closer to the language bit on the left-side ^rule1, which does the important job of comprehension for you ^rule3 (ie, what does she mean by the words 'crash' and 'car'?), and also is the side which applies the logic to what you are hearing (ie, 'Has she phoned the AA?' and 'What about the no-claims bonus?') Using the right side of your brain is unnecessary ^rule5, because comprehending and applying logic is all that matters.

Right-handed women

If you are a right-handed woman, the chances are you will also be using your right ear ^rule11 even though it isn't quite as good at listening as your left ear ^rule2. When the other half mentions he's crashed the car you will also swap to the left ear ^rule1, because the right-side of your brain is already now being used to apply the emotion to what you are hearing ^rule4 (ie, 'Is he all right?' and 'Is he really all right?'), and your left-side is ready to do a bit more work on the comprehension ^rule6 just in case there are some important details to be thought through.

Left-handers

Those of us unfortunate enough to be left-handed will already have the phone to the left ear ^rule12. There is no need to swap ^rule1. We let the left-side of our brain process the language. For left-handed men that's all there is to it ^rule5. For left-handed women, the right-side of the brain is still doing the emotional application at the same time ^rule6.

To Swap or Not to Swap

Right-handed people have learned to use the phone with their right ear, but when it comes to the crunch, they swap to the left ear. Left-handed people have the advantage of using the 'correct' ear for listening with, but we haven't spent our entire lives retraining our brain to listen with the right ear at all. This is fine for us until we want to write with our left hand while on the phone using our left ear, at which point have to either:
(a) swap the phone to the right ear in order to be able to write coherently, but then suddenly we can't hear a thing,
or,
(b) jam the phone between ear and shoulder to be able to hear, but then we can't write very well.

Deer and Rabbits

Thousands of years of natural selection has ensured that not many left-handers remain:

In times past our fathers would be out hunting, using the better suited left-sides of their brains to do a good job of throwing spears and the like with their right hands. Those left-handed hunters wouldn't have been quite as good a shot, and so would come home with less food. While our left-handed dads were out chasing deer, our mums were cradling us in their left arms - not because they were left-handed, but because our mothers were better at reading our mood when we were on their left, as the sensory input would reach the emotional right-side of their brains first. This also meant that they had their right hands free for throwing stones at rabbits just in case our dads didn't come back with particularly large bits of venison.

Phones and Post-its

We've moved on from those Neanderthal days. However, the phone is now the instrument of our continuing demise:

Our fathers are now out working, sitting at desks, answering the phones and writing messages on Post-it notes. Those left-handed dads are either not quite hearing what has been said on the phone, or are writing incoherent messages on Post-its for actioning after lunch. Either way, they are going to get passed over for that promotion, and will therefore come home with less food. Our mothers are still cradling us in their left arms, for the same reasons as their ancestors, but they are keeping their right hands free for answering the phone to our dads when they call to say that they got passed over for that promotion again, so had one to many consolation beers at lunchtime, and have consequently crashed the car on the way home.